Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Paths in Weyl chambers and random matrices

Auteur(s):

Code(s) de Classification MSC:

• 15A52 Random matrices
• 17B10 Representations, algebraic theory (weights)
• 60B99 None of the above but in this section
Résumé: Baryshnikov [3] and Gravner, Tracy \& Widom [14] have shown that the largest eigenvalue of a random matrix of the G.U.E. of order $d$ has the same distribution as $$\max_{1 \geq t_1 \geq \cdots \geq t_{d-1}\geq 0}\left[ W_1(1)-W_1(t_{1}) +W_2(t_1)-W_2(t_2)+\cdots + W_d(t_{d-1})\right],$$ where $W=(W_1,\cdots,W_d)$ is a $d$-dimensional Brownian motion. We provide a generalization of this formula to all the eigenvalues and give a geometric interpretation. For any Weyl chamber $\cone$ of an Euclidean finite-dimensional space $\a$, we define a natural continuous path transformation $\T$ which associates to a path $\w$ in $\a$ a path $\T \w$ in $\conec$. This transformation occurs in the description of the asymptotic behaviour of some deterministic dynamical systems on the symmetric space $G/K$ where $G$ is the complex group with chamber $\cone$. When $\a=\R^d$, $\cone=\{(x_1,\cdots,x_d);x_1>x_2>\cdots >x_d\}$ and if $W$ is the Euclidean Brownian motion on $\a$ then $\T W$ is the process of the eigenvalues of the Dyson Brownian motion on the set of Hermitian matrices and $(\T W)(1)$ is distributed as the eigenvalues of the G.U.E.